On the Wecken property for the root problem of mappings between surfaces

Let $M_1$ and $M_2$ be two closed (not necessarily orientable) surfaces, $f\colon M_1\to M_2$ be a continuous map, and $c$ be a point in $M_2$. By definition, the map $f$ has the Wecken property for the root problem if $f$ can be deformed into a map $\tilde f$ such that the number $|\tilde f{-1}(c)|$ of roots of $\tilde f$ coincides with the number ${\rm NR}[f]$ of the essential Nielsen root classes of $f$, that is, ${\rm MR}[f]={\rm NR}[f]$. We characterize the pairs of surfaces $M_1$, $M_2$ for which all continuous mappings $f\colon M_1\to M_2$ have the Wecken property for the root problem. The criterion is formulated in terms of the Euler characteristics of the surfaces and their orientability properties.